The relativistic spherical \(\delta\)-shell interaction in \(\mathbb{R}^3\): spectrum and approximation. (English) Zbl 1370.81071
Summary: This note revolves on the free Dirac operator in \(\mathbb{R}^3\) and its \(\delta\)-shell interaction with electrostatic potentials supported on a sphere. On one hand, we characterize the eigenstates of those couplings by finding sharp constants and minimizers of some precise inequalities related to an uncertainty principle. On the other hand, we prove that the domains given by J. Dittrich et al. [ibid. 30, No. 12, 2875–2882 (1989; Zbl 0694.46053)] and by N. Arrizabalaga et al. [J. Math. Pures Appl. (9) 102, No. 4, 617–639 (2014; Zbl 1297.81083)] for the realization of an electrostatic spherical shell interaction coincide. Finally, we explore the spectral relation between the shell interaction and its approximation by short range potentials with shrinking support, improving previous results in the spherical case.{
©2017 American Institute of Physics}
©2017 American Institute of Physics}
MSC:
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 81R20 | Covariant wave equations in quantum theory, relativistic quantum mechanics |
| 78A30 | Electro- and magnetostatics |
| 47B25 | Linear symmetric and selfadjoint operators (unbounded) |
| 35Q40 | PDEs in connection with quantum mechanics |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 62J10 | Analysis of variance and covariance (ANOVA) |
Software:
DLMFReferences:
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